356 FIGURES OF EQUILIBRIUM OF A LIQUID MASS 
the maximum height of the partial catenoid is, cither exactly or very nearly, 3 
of the diameter of the bases. - This catenoid is represented by Fig. 21. 
Let us now call attention to the slight spontaneous transformations, considered 
in themselves. Till now, when we saw one of our liquid figures become trans- 
formed, and thus pass from an unstable to a stable equilibrium, the alteration 
was profound, the mass separated into two or several parts, and the final result 
of the phenomenon always consisted of spheres or portions of spheres. Here, 
there is nothing of the kind: the alteration is inconsiderable; the mass does not 
disunite, and the final result is a figure which deviates little from the former, 
at least in the portion realized, and which may be of the same nature. In the 
first experiment, for example, an unstable partial unduloid is transformed into 
another unduloid but little different, and doubtless the same is the case in the 
second. Moreover, what is still more remarkable, the comparison of the first 
two experiments seems to indicate that the unstable unduloid and the stable 
unduloid into which it is converted approach one another indefinitely in pro- 
portion as the distance of the rings is nearer the maximum height of the cate- 
noid. 
The experiments which we are discussing furnish the key of the difficulty 
indicated at the end of § 18 in regard to the stability of the partial catenoid of 
greatest height. When, the rings being at the distance which corresponds to 
this catenoid and a cylinder formed between them, the small syringe is put in 
operation, the figure becomes, as we know, unduloid, which, varying with the 
progress of the absorption, tends towards the catenoid; but the third experi- 
ment further shows that if, after having attained that limit, we continue the ope- 
ration, the figure again insensibly becomes an unduloid which deviates, in pro- 
portion to the exhaustion, from this same catenoid. If, then, the partial catenoid 
of greatest height constitutes the transition between partial catenoids stable and 
partial catenoids unstable, it constitutes, on the other hand, the transition be- 
tween a continuous series of stable unduloids and another continuous series of 
unduloids equally stable. Such is evidently the reason of the decided stability 
of the partial catenoid of greatest height; hence, when, by means which will 
be explained in a subsequent series, we render impossible the formation of every 
other figure but the catenoid, this loses its stability as soon as we give it the 
maximum height. 
We close here the study of the unduloid and catenoid and pass to that of a 
third figure. 
§ 22. Of this third figure we already know a portion: it is the constriction 
with concave bases obtained in the jast two experiments of § 20, a constriction 
which, by the nature of those bases, is foreign to the unduloid and catenoid. 
To realize it, it is requisite, as has been seen, that the distance of the rings 
should be less than % of the diameter; Fig. 22 represents, in its meridian sec- 
tion, such a constriction, for a distance of the rings equal to about a third of the 
diameter, and when the bases have already become strongly concave; the dot- 
ted lines are sections of the planes of the rings. Let us now endeavor, as in 
